| L(s) = 1 | + 4.58·2-s + 12.9·4-s + 12.2·5-s + 15.3·7-s + 22.8·8-s + 56.3·10-s + 41.7·11-s − 23.4·13-s + 70.3·14-s + 0.870·16-s − 123.·17-s + 106.·19-s + 159.·20-s + 191.·22-s + 207.·23-s + 26.2·25-s − 107.·26-s + 199.·28-s + 53.4·29-s − 252.·31-s − 179.·32-s − 567.·34-s + 188.·35-s − 357.·37-s + 486.·38-s + 281.·40-s + 353.·41-s + ⋯ |
| L(s) = 1 | + 1.61·2-s + 1.62·4-s + 1.09·5-s + 0.828·7-s + 1.01·8-s + 1.78·10-s + 1.14·11-s − 0.501·13-s + 1.34·14-s + 0.0135·16-s − 1.76·17-s + 1.28·19-s + 1.78·20-s + 1.85·22-s + 1.87·23-s + 0.209·25-s − 0.812·26-s + 1.34·28-s + 0.341·29-s − 1.46·31-s − 0.988·32-s − 2.86·34-s + 0.911·35-s − 1.58·37-s + 2.07·38-s + 1.11·40-s + 1.34·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.723992587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.723992587\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 - 59T \) |
| good | 2 | \( 1 - 4.58T + 8T^{2} \) |
| 5 | \( 1 - 12.2T + 125T^{2} \) |
| 7 | \( 1 - 15.3T + 343T^{2} \) |
| 11 | \( 1 - 41.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 23.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 123.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 106.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 207.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 53.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 252.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 357.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 353.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 80.6T + 7.95e4T^{2} \) |
| 47 | \( 1 - 5.12T + 1.03e5T^{2} \) |
| 53 | \( 1 - 260.T + 1.48e5T^{2} \) |
| 61 | \( 1 - 35.7T + 2.26e5T^{2} \) |
| 67 | \( 1 - 635.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 644.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 531.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 594.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 95.7T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.00e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 964.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.92858851587874992497109607212, −9.473600617988797681034404769301, −8.866438068480182693955055474371, −7.16074059017523745421909967325, −6.59534363472456607638175252820, −5.44417471635253854411032351657, −4.93024411342543523123053300551, −3.84519516123026311583525251719, −2.56002903790241846079486250359, −1.55366107008990883321066841268,
1.55366107008990883321066841268, 2.56002903790241846079486250359, 3.84519516123026311583525251719, 4.93024411342543523123053300551, 5.44417471635253854411032351657, 6.59534363472456607638175252820, 7.16074059017523745421909967325, 8.866438068480182693955055474371, 9.473600617988797681034404769301, 10.92858851587874992497109607212
